Term Rewriting System R:
[x, y, z]
+(*(x, y), *(x, z)) -> *(x, +(y, z))
+(+(x, y), z) -> +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

+(*(x, y), *(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

where the Polynomial interpretation:
  POL(*(x1, x2))=  1 + x1 + x2  
  POL(+(x1, x2))=  x1 + x2  
  POL(u)=  0  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

+(+(x, y), z) -> +(x, +(y, z))

where the Polynomial interpretation:
  POL(+(x1, x2))=  1 + 2·x1 + x2  
was used.

All Rules of R can be deleted. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳OC
             ...
               →TRS4
                 ↳Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

Termination of R successfully shown.
Duration:
0:00 minutes